Active elastohydrodynamics of vesicles in narrow blind constrictions

Fluid-resistance limited transport of vesicles through narrow constrictions is a recurring theme in many biological and engineering applications. Inspired by the motor-driven movement of soft membrane-bound vesicles into closed neuronal dendritic spines, here we study this problem using a combination of passive three-dimensional simulations and a simplified semianalytical theory for the active transport of vesicles forced through constrictions by molecular motors. We show that the motion of these objects is characterized by two dimensionless quantities related to the geometry and to the strength of forcing relative to the vesicle elasticity. We use numerical simulations to characterize the transit time for a vesicle forced by fluid pressure through a constriction in a channel and find that relative to an open channel, transport into a blind end leads to the formation of a smaller forward-flowing lubrication layer that strongly impedes motion. When the fluid pressure forcing is complemented by forces due to molecular motors that are responsible for vesicle trafficking into dendritic spines, we find that the competition between motor forcing and fluid drag results in multistable dynamics reminiscent of the real system. Our study highlights the role of nonlocal hydrodynamic effects in determining the kinetics of vesicular transport in constricted geometries.

Figure 1

(a) Time-lapse images and (b) corresponding kymograph from a dual-color time-lapse recording showing the actomyosin-based transportation of an endosome through the neck of a neuronal dendritic spine over 40–60 s (adapted from [1] and available under CC BY NC ND license). The bright color represents the endosome and the curve in (a) indicates the contour of the spine. The region of high intensity at the base of the spine represents a pool of endosomes, from which the kymograph shows one entrance event. (c) Our corresponding model system for the forced translocation of a vesicle through a narrow constriction. (d) Schematic of the motor model used, which includes two species of motors, one pushing up the channel and another pushing down the channel.

Figure 2

Translocation sequence and color map of the $z$ velocity in the $x\text{−}z$ plane through (a) closed and (b) open constrictions for a vesicle with radius ${\pi }_1=0.75$ and applied dimensionless forcing ${\pi }_2=0.13$. While the shape of the vesicles during the translocation sequence is not markedly different, the associated velocities differ significantly. The color map in (a) of the $z$ velocity in the $x\text{−}z$ plane indicates the occurrence of a narrow fluid layer with reverse flow in the closed constriction that does not occur in the open constriction. (c) Time evolution of the dimensionless position of the center of mass of the vesicle $Z/R_c$ passing through open (red) and closed (black) constrictions with otherwise identical parameters. Note that the velocity of the vesicle through the open constriction is only 5% slower in the narrow part of the channel, whereas for the closed constriction it is 5 times slower. (d) Fluid $z$ velocity $v_z$ in the closed constriction relative to the vesicle velocity $U$ as a function of $r/R_c$, the radial coordinate $r$ relative to the radius of the constriction, for applied forces $F=3$, 4, and 5 corresponding to ${\pi }_2=0.16$, 0.21, and 0.27 respectively. The data collapses in dimensionless coordinates, as evidenced by comparison to the inset, which shows dimensional velocities at different forcing strengths.

Figure 3

(a) Relative minimal gap size $h_0/R_c$ as a function of ${\pi }_1=R_p/R_c$ for $F=2.5$ and 5, where $h_0$ is defined to be the minimal lubrication gap during translocation (see the inset). (b) Relative transit time $\tau /{\tau }_0$ as a function of $R_p/R_c$ for $F=2.5$ and 5. Plotting the same data on a log-log scale indicates that $\tau /{\tau }_0\sim {(h_0/R_c)}^{-2}$ for gap sizes $0.05<h_0/R_c<0.5$ (inset), though we cannot definitively establish a power-law behavior from the full 3D simulations since the gap sizes range only over a single decade. Here the vesicle stiffness is kept constant in all simulations and the corresponding dimensionless forcing is ${\pi }_2=0.1–0.5$.

Figure 4

(a) The trajectory obtained from the lubrication model (LM) is in good agreement with the 3D LB simulation with parameters from Tables 1 and 2 such that ${\pi }_1=0.83$ and ${\pi }_2=0.090$. The discrepancy in the wide part of the channel is not surprising, since the assumptions of the lubrication model break down when $h\sim R_p$. (b) Snapshots during translocation from the 3D lattice Boltzmann simulations. (c) The lubrication model was used to investigate the scaling of the nondimensional transit time $\tau /{\tau }_0$ with the relative vesicle size $1-{\pi }_1\approx h_0/R_c$ and the normalized force ${\pi }_2$. (d) Transit time versus $1-{\pi }_1$ on a logarithmic scale in the limit ${\pi }_1\to 1$. According to the scaling argument in Sec. 3a, $\tau \sim h_0^{-5/2}$ in the inelastic regime ${\pi }_2/2\ll 1-{\pi }_1$ (blue line). The lubrication model recovers this limit in the inelastic case (blue symbols). In the elastic case with ${\pi }_2=1.8\times {10}^{-3}$ (red symbols), solving the lubrication model reveals a weaker dependence on the height and a plateau with minimum height $h_0/R_p\approx {\pi }_2/2$ at the transition ${log}_{10}(1-{\pi }_1)={log}_{10}({\pi }_2/2)\approx -3.0$.

Figure 5

Bifurcation analysis of competing motor species pushing a rigid vesicle through a constriction into a blind end. (a) Force-velocity curves for a single motor species forming connections at $z=-A$ (blue dashed line) and a single motor species forming connections at $z=A$ (red dash-dotted line) along with the sum of the two force-velocity curves (black solid line). We have set the value of the dimensionless parameter ${\pi }_5:=\gamma (B-A)$ to be ${\pi }_5=0.1$. (b) Steady-state solutions obtained by plotting the difference between the motor force-velocity curve and the viscous drag $F=\zeta U$ for friction coefficient $\zeta \approx 4.5\times {10}^{-8}$ kg/s. For reference, the friction coefficient in free space predicted by Stokes' law is $\zeta \approx 2.7\times {10}^{-8}$ kg/s. Arrows are drawn to illustrate the effect of small perturbations from steady states; e.g. perturbations from the leftmost steady state result in acceleration back toward the leftmost steady state, hence it is stable (closed circles) as opposed to the two unstable steady states (open circles) from which small perturbations give rise to divergent trajectories. The inset shows a close-up of the three steady states near the origin. (c) Bifurcation diagram showing the steady-state velocities as a function of the fraction of motors ${\varphi }_1$ pushing up the channel. Solid lines are used to denote stable steady states, while dashed lines are used for unstable steady states. (d) Bifurcation diagram showing the dependence of the steady-state velocities on the friction coefficient $\zeta$, under the assumption of equal fractions of motors pushing in each direction $\left({\varphi }_1={\varphi }_2=0.5\right)$.

Figure 6

Vesicle trajectory and velocity resulting from different fractions ${\varphi }_1$ of upward-directed motors. (a) The vesicle travels in a processive fashion to the tip of the spine with ${\varphi }_1=0.57,R_p=0.96\mu \mathrm{m},R_c=1.22\mu \mathrm{m}$, and $F_0=50$ pN. (b) The vesicle becomes trapped in the spine neck as it encounters a bifurcation in the steady-state solutions when ${\varphi }_1=0.5$ and with all other parameters unchanged. (c) Demonstration of multistability in the lubrication model; both positive and negative velocities occur as in experiment [1] when using ${\varphi }_1=0.46,R_p=1.5\mu \mathrm{m},R_c=2.15\mu \mathrm{m},F_0=200$ pN, and ${\pi }_5=0.02$. (d) Multistability is obtained using the opposite motor polarity with ${\varphi }_1=0.54,R_p=1.5\mu \mathrm{m},R_c=2.15\mu \mathrm{m},F_0=200$ pN, and ${\pi }_5=0.02$.

Figure 7

(a) Velocity of the vesicle as a function of position at two different grid resolutions and otherwise identical parameters. (b) Velocity of the vesicle as a function of position at three different mesh sizes. (c) Color map of the $z$ velocity in the $x\text{−}y$ plane showing the radially symmetric velocity profile halfway through the constriction at $z=180$. (d) Position of the center of mass of the vesicle as a function of time for identical dimensionless groups ${\pi }_1=0.73$ and ${\pi }_2=0.27$. Here the black curve corresponds to the moduli of Table 1 and the red curve corresponds to elastic moduli and applied force one order of magnitude smaller. The inset shows that the resulting rescaled velocities in the constriction are nearly identical.